Problem: You have found the following ages (in years) of all 6 sloths at your local zoo: $ 1,\enspace 22,\enspace 9,\enspace 10,\enspace 11,\enspace 12$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we have data for all 6 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{1 + 22 + 9 + 10 + 11 + 12}{{6}} = {10.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $1$ year $-9.8$ years $96.04$ years $^2$ $22$ years $11.2$ years $125.44$ years $^2$ $9$ years $-1.8$ years $3.24$ years $^2$ $10$ years $-0.8$ years $0.64$ years $^2$ $11$ years $0.2$ years $0.04$ years $^2$ $12$ years $1.2$ years $1.44$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{96.04} + {125.44} + {3.24} + {0.64} + {0.04} + {1.44}} {{6}} $ $ {\sigma^2} = \dfrac{{226.84}}{{6}} = {37.81\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{37.81\text{ years}^2}} = {6.1\text{ years}} $ The average sloth at the zoo is 10.8 years old. There is a standard deviation of 6.1 years.